A particle of mass $m$ moves under the influence of a central force given by $F(r) = -k/r^3$. If the particle's orbit is circular with radius $r_0$, its time period of revolution is:

A block of mass 'm' is sliding down an inclined plane at an angle $heta$ with the horizontal. Derive an expression for the acceleration of the block, assuming no friction.

The dimensions of ${e^2}/4{\rm{\pi }}{{\rm{\varepsilon }}_0}hc,$, where $e,\;{{\rm{\varepsilon }}_0},{\rm{\;}}h\;{\rm{and\;}}c$ are electronic charge, electric permittivity, Planck’s constant and velocity of light in vacuum respectively, are

One yard in SI units is equal

The dimensional formula of coefficient of permittivity for free space $\left( {{\varepsilon _0}} \right)$ in the equation $F = \frac{1}{{4{\rm{\pi }}{\varepsilon _0}}}\frac{{{q_1}{q_2}}}{{{r^2}}},$ where symbols have their usual meanings, is

A particle is projected with velocity $v_0$ at an angle $heta$ with the horizontal. At the highest point of its trajectory, it explodes into two fragments of masses $m_1$ and $m_2$. The lighter fragment $m_1$ falls vertically downwards with zero initial speed. If the range of the original projectile was $R$, the distance from the point of projection where the heavier fragment lands is:

A screw gauge has a pitch of 1 mm and 100 divisions on its circular scale. The zero of the circular scale lies 6 divisions below the mean line when the two studs of the screw gauge are in contact. What is the zero error of this screw gauge?

Which equation is derived from the definition of acceleration?

The velocity v of water waves may depend on their wavelength ($\lambda$), the density of water ($\rho$) and the acceleration due to gravity (g). The method of dimensions gives the relation between these quantities as

If the speed of light (c), acceleration due to gravity (g) and pressure (p) are taken as the fundamental quantities, then the dimension of gravitational constant is

A quantity X is given by ${\varepsilon _0}L\frac{{\Delta V}}{{\Delta t}},\;$, where ${\varepsilon _0}$ is the permittivity of free space, L is a length, ${\Delta V}$ is a potential difference and ${\Delta t}$ is a time interval. The dimensional formula for X is the same as that of